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Sensitivity analysis for nonlinear hyperbolic systems of conservation laws
LMV, UVSQ, Versailles Université Côte d’Azur, INRIA, Sophia-Antipolis
Camilla Fiorini1
Christophe Chalons1
Régis Duvigneau 2
1 2
Under the supervision of
11 / 07 / 2018
PhD defence
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Outline of the talk
‣ Sensitivity analysis
‣ Sensitivity analysis for hyperbolic equations
‣ Riemann problem for the Euler equations and their sensitivity
‣ Classical numerical schemes
‣ Anti-diffusive numerical schemes
‣ Applications
1
SensitivityAnalysis
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Sensitivity Analysis
Sensitivity analysis: study of how changes in the inputs of a model affect the outputs
inputparameters
output(state)model
a U
Sensitivity: @U
@a= Ua
2
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Applications
‣ Optimization [5]
3
[5] Borggaard, J., & Burns, J. (1997). A PDE sensitivity equation method for optimal aerodynamic design. Journal of Computational Physics, 136(2), 366-384.[6] Duvigneau, R., & Pelletier, D. (2006). A sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parameters. International Journal of Computational Fluid Dynamics, 20(7), 497-512.[7] Delenne, C. (2014). Propagation de la sensibilité dans les modèles hydrodynamiques (HDR, Montpellier II).
@J(U)
@a= b(U,Ua)
mina2A
J(U)Problem: , where
Classical optimization techniques call for the differentiation of the cost function:
J(U) =1
2b(U,U) and is bilinear.b
anew = aold � ↵@J(U)
@a
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Applications
‣ Optimization [5]
3
[5] Borggaard, J., & Burns, J. (1997). A PDE sensitivity equation method for optimal aerodynamic design. Journal of Computational Physics, 136(2), 366-384.[6] Duvigneau, R., & Pelletier, D. (2006). A sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parameters. International Journal of Computational Fluid Dynamics, 20(7), 497-512.[7] Delenne, C. (2014). Propagation de la sensibilité dans les modèles hydrodynamiques (HDR, Montpellier II).
‣ Quick evaluation of close solutions [6]
U(a+ �a) = U(a) + �aUa(a) + o(�a2)
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Applications
‣ Optimization [5]
3
[5] Borggaard, J., & Burns, J. (1997). A PDE sensitivity equation method for optimal aerodynamic design. Journal of Computational Physics, 136(2), 366-384.[6] Duvigneau, R., & Pelletier, D. (2006). A sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parameters. International Journal of Computational Fluid Dynamics, 20(7), 497-512.[7] Delenne, C. (2014). Propagation de la sensibilité dans les modèles hydrodynamiques (HDR, Montpellier II).
‣ Quick evaluation of close solutions [6]
‣ Uncertainty quantification [7]
µ U(µa)
�2 Ua(µa)TUa(µa)�
2a
First order estimates
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Two approaches
4
analytical state model
analytical sensitivity model
discrete sensitivity model
differentiate
discretise
discrete state model
analytical state model
discrete sensitivity model
discretise
differentiate
Differentiate then discretiseDiscretise then differentiate
analytical sensitivity model
no discretisation of computational
facilitators
could lead to incoherent gradients
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Standard techniques of sensitivity analysis call for the differentiation of the state system:
Standard techniques of sensitivity analysis
(@
t
U+ @
x
F(U) = 0 ⌦⇥ (0, T ),
U(x, 0) = g(x) ⌦,
5
The global system, without the correction term, is only weakly hyperbolic.
The matrix of the global system has the following structure:
[8] Bardos, C., & Pironneau, O. (2002). A formalism for the differentiation of conservation laws. Comptes Rendus Mathematique, 335(10), 839-845.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Standard techniques of sensitivity analysis call for the differentiation of the state system:
Standard techniques of sensitivity analysis
(@
a
(@t
U) + @
a
(@x
F(U)) = 0 ⌦⇥ (0, T ),
@
a
U(x, 0) = @
a
g(x) ⌦,
5
The global system, without the correction term, is only weakly hyperbolic.
The matrix of the global system has the following structure:
[8] Bardos, C., & Pironneau, O. (2002). A formalism for the differentiation of conservation laws. Comptes Rendus Mathematique, 335(10), 839-845.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Standard techniques of sensitivity analysis call for the differentiation of the state system:
Standard techniques of sensitivity analysis
(@
a
(@t
U) + @
a
(@x
F(U)) = 0 ⌦⇥ (0, T ),
@
a
U(x, 0) = @
a
g(x) ⌦,
5
The global system, without the correction term, is only weakly hyperbolic.
The matrix of the global system has the following structure:
[8] Bardos, C., & Pironneau, O. (2002). A formalism for the differentiation of conservation laws. Comptes Rendus Mathematique, 335(10), 839-845.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Standard techniques of sensitivity analysis call for the differentiation of the state system:
Standard techniques of sensitivity analysis
(@
t
Ua
+ @
x
Fa
(U,Ua
) = 0 ⌦⇥ (0, T ),
Ua
(x, 0) = ga
(x) ⌦,
5
The global system, without the correction term, is only weakly hyperbolic.
The matrix of the global system has the following structure:
[8] Bardos, C., & Pironneau, O. (2002). A formalism for the differentiation of conservation laws. Comptes Rendus Mathematique, 335(10), 839-845.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Standard techniques of sensitivity analysis call for the differentiation of the state system:
This can be done under hypotheses of regularity of the state U [8].
Standard techniques of sensitivity analysis
If these techniques are applied to hyperbolic equations, Dirac delta functions will appear in the sensitivity.
(@
t
Ua
+ @
x
Fa
(U,Ua
) = 0 ⌦⇥ (0, T ),
Ua
(x, 0) = ga
(x) ⌦,
5
For the Burgers’ equation:
F(U) =u2
2Fa(U,Ua) = uua
The global system, without the correction term, is only weakly hyperbolic.
The matrix of the global system has the following structure:
[8] Bardos, C., & Pironneau, O. (2002). A formalism for the differentiation of conservation laws. Comptes Rendus Mathematique, 335(10), 839-845.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Standard techniques of sensitivity analysis
6
From different point of view: the global system is weakly hyperbolic.
A 0B A
�
therefore it has repeated eigenvalues. We proved that in the general case it is not diagonalisable.
matrix of the state system
The Jacobian matrix of the global system has the following form:
For the Burgers’ equation:u 0ua u
�Jacobian matrix:
Sensitivity analysisfor hyperbolic
equations
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
In order to have a sensitivity system which is valid also when the state is discontinuous, we add a correction term [9]:
Definition of the source term
S =NsX
k=1
⇢
⇢
⇢k�(x� xk,s(t)),
(@
t
Ua
+ @
x
Fa
(U,Ua
) = S ⌦⇥ (0, T ),
Ua
(x, 0) = ga
(x) ⌦,
defined as follows:number of discontinuities
position of the k-th discontinuity
amplitude of the k-th correction (to be computed)
Remark: a shock detector is necessary to discretise such source term.
7
[9] Guinot, V., Delenne, C., & Cappelaere, B. (2009). An approximate Riemann solver for sensitivity equations with discontinuous solutions. Advances in Water Resources, 32(1), 61-77.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Definition of the source term
x1 x2xc
t1
t2
�
To compute the amplitude of the correction, we consider an infinitesimal control volume containing a single discontinuity:
By integrating the sensitivity equations with the source term on the control volume, one has:
Rankine-Hugoniot conditions for the state:
Differentiating them w.r.t. the parameter:
Finally, we obtain the following amplitude:
⇢⇢⇢k = (U�a �U+
a )�k + F+a � F�
a
(U+ �U�)�k = F+ � F�
8
(U+a
�U�a
)�k
+ (U+ �U�)�k,a
+ �
k
(@x
U+ � @
x
U�)@a
x
k,s
(t) =
= F+a
� F�a
+
✓@F(U+)
@U@
x
U+ � @F(U�)
@U@
x
U�◆@
a
x
k,s
(t).
⇢
⇢
⇢
k
= (U+ �U�)�k,a
+ �
k
(@x
U+ � @
x
U�)@a
x
k,s
(t)�✓@F(U+)
@U@
x
U+ � @F(U�)
@U@
x
U�◆@
a
x
k,s
(t).
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Thesis synopsis
‣ Chapter 1: introduction
‣ Chapter 2: scalar case
‣ Chapter 3: the p-system [1,2]
‣ Chapter 4: the Euler system [3]
‣ Chapter 5: quasi 1D Euler system
‣ Chapter 6: conclusion and perspectives
‣ Appendix A: modelling of running strategies [4]
9
[1] Chalons, C., Duvigneau, R., & Fiorini, C. (2017). Sensitivity analysis for the Euler equations in Lagrangian coordinates. In International Conference on Finite Volumes for Complex Applications (pp. 71-79). Springer, Cham.[2] Chalons, C., Duvigneau, R. & Fiorini, C. (2017). Sensitivity analysis and numerical diffusion effects for hyperbolic PDE systems with discontinuous solutions. The case of barotropic Euler equations in Lagrangian coordinates. Submitted to SJSC.[3] Fiorini, C., Chalons, C., Duvigneau, R (2018). Sensitivity equation method for Euler equations in presence of shocks applied to uncertainty quantification. Submitted to JCP.[4] Fiorini, C. (2017). Optimization of Running Strategies According to the Physiological Parameters for a Two-Runner Model. Bulletin of mathematical biology, 79(1), 143-162.
Riemann problemfor Euler equationsand their sensitivity
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
The Riemann problem for Euler equations
The Euler equations write:8<
:
@t
⇢+ @x
(⇢u) = 0,@t
(⇢u) + @x
(⇢u2 + p) = 0,@t
(⇢E) + @x
(u(⇢E + p)) = 0,
�1(U) = u� c,�2(U) = u,�3(U) = u+ c.
r1(U) = (1, u� c,H � uc)t,
r2(U) = (1, u, u2
2 )t,r3(U) = (1, u+ c,H + uc)t.
Eigenvalues: Eigenvectors:
Genuinely nonlinear
\begin{tikzpicture}[scale=1]\draw[->] (0,0) -- (5,0);\draw[->] (0,0) -- (0,2.5);\draw (2.5,-.1) -- (2.5, .1);\draw[blue] (2.5,0) -- (0.9,2.5);\draw[blue] (2.5,0) -- (1.1,2.5);\draw[blue] (2.5,0) -- (1.3,2.5);\draw[blue] (2.5,0) -- (1.5,2.5);\draw[blue] (2.5,0) -- (1.7,2.5);\draw[red, dashed, thick] (2.5,0) -- (2.7,2.5);\draw[red] (2.5,0) -- (4.2,2.5);\node[below] at (5,-.1) {$x$};\node[left] at (-.1,2.5) {$t$};\node[below] at (2.5,-.1) {$x_c$};\node[below] at (1.25,1.25) {$\mathbf{U}_L$};\node[below] at (2.3,2.2) {$\mathbf{U}^*_L$};\node[below] at (3.2,2.2) {$\mathbf{U}^*_R$};\node[below] at (3.75,1.25) {$\mathbf{U}_R$};\node[below] at (1.6,2.2) {$\widehat{\mathbf{U}}$};
\end{tikzpicture}
\begin{tikzpicture}[scale=1] \draw[->] (0,0) -- (5,0); \draw[->] (0,0) -- (0,2.5); \draw (2.5,-.1) -- (2.5, .1); \draw[red, dashed, thick] (2.5,0) -- (2.7,2.5); \draw[red] (2.5,0) -- (4.2,2.5); \node[below] at (2.3,2.2) {$\mathbf{U}^*_L$}; \node[below] at (3.2,2.2) {$\mathbf{U}^*_R$}; \node[below] at (3.75,1.25) {$\mathbf{U}_R$}; \draw[->] (0,0) -- (5,0); \draw[->] (0,0) -- (0,2.5); \draw (2.5,-.1) -- (2.5, .1); \draw[red, dashed, thick] (2.5,0) -- (2.7,2.5); \draw[red] (2.5,0) -- (1.7,2.5); \node[below] at (5,-.1) {$x$};
10
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
The Riemann problem for Euler equations
The Euler equations write:8<
:
@t
⇢+ @x
(⇢u) = 0,@t
(⇢u) + @x
(⇢u2 + p) = 0,@t
(⇢E) + @x
(u(⇢E + p)) = 0,
�1(U) = u� c,�2(U) = u,�3(U) = u+ c.
r1(U) = (1, u� c,H � uc)t,
r2(U) = (1, u, u2
2 )t,r3(U) = (1, u+ c,H + uc)t.
Eigenvalues: Eigenvectors:
Linearly degenerate
\begin{tikzpicture}[scale=1]\draw[->] (0,0) -- (5,0);\draw[->] (0,0) -- (0,2.5);\draw (2.5,-.1) -- (2.5, .1);\draw[blue] (2.5,0) -- (0.9,2.5);\draw[blue] (2.5,0) -- (1.1,2.5);\draw[blue] (2.5,0) -- (1.3,2.5);\draw[blue] (2.5,0) -- (1.5,2.5);\draw[blue] (2.5,0) -- (1.7,2.5);\draw[red, dashed, thick] (2.5,0) -- (2.7,2.5);\draw[red] (2.5,0) -- (4.2,2.5);\node[below] at (5,-.1) {$x$};\node[left] at (-.1,2.5) {$t$};\node[below] at (2.5,-.1) {$x_c$};\node[below] at (1.25,1.25) {$\mathbf{U}_L$};\node[below] at (2.3,2.2) {$\mathbf{U}^*_L$};\node[below] at (3.2,2.2) {$\mathbf{U}^*_R$};\node[below] at (3.75,1.25) {$\mathbf{U}_R$};\node[below] at (1.6,2.2) {$\widehat{\mathbf{U}}$};
\end{tikzpicture}
\begin{tikzpicture}[scale=1] \draw[->] (0,0) -- (5,0); \draw[->] (0,0) -- (0,2.5); \draw (2.5,-.1) -- (2.5, .1); \draw[red, dashed, thick] (2.5,0) -- (2.7,2.5); \draw[red] (2.5,0) -- (4.2,2.5); \node[below] at (2.3,2.2) {$\mathbf{U}^*_L$}; \node[below] at (3.2,2.2) {$\mathbf{U}^*_R$}; \node[below] at (3.75,1.25) {$\mathbf{U}_R$}; \draw[->] (0,0) -- (5,0); \draw[->] (0,0) -- (0,2.5); \draw (2.5,-.1) -- (2.5, .1); \draw[red, dashed, thick] (2.5,0) -- (2.7,2.5); \draw[red] (2.5,0) -- (1.7,2.5); \node[below] at (5,-.1) {$x$};
10
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
The Riemann problem for Euler equations
The Euler equations write:8<
:
@t
⇢+ @x
(⇢u) = 0,@t
(⇢u) + @x
(⇢u2 + p) = 0,@t
(⇢E) + @x
(u(⇢E + p)) = 0,
�1(U) = u� c,�2(U) = u,�3(U) = u+ c.
r1(U) = (1, u� c,H � uc)t,
r2(U) = (1, u, u2
2 )t,r3(U) = (1, u+ c,H + uc)t.
Eigenvalues: Eigenvectors:\begin{tikzpicture}[scale=1]
\draw[->] (0,0) -- (5,0);\draw[->] (0,0) -- (0,2.5);\draw (2.5,-.1) -- (2.5, .1);\draw[blue] (2.5,0) -- (0.9,2.5);\draw[blue] (2.5,0) -- (1.1,2.5);\draw[blue] (2.5,0) -- (1.3,2.5);\draw[blue] (2.5,0) -- (1.5,2.5);\draw[blue] (2.5,0) -- (1.7,2.5);\draw[red, dashed, thick] (2.5,0) -- (2.7,2.5);\draw[red] (2.5,0) -- (4.2,2.5);\node[below] at (5,-.1) {$x$};\node[left] at (-.1,2.5) {$t$};\node[below] at (2.5,-.1) {$x_c$};\node[below] at (1.25,1.25) {$\mathbf{U}_L$};\node[below] at (2.3,2.2) {$\mathbf{U}^*_L$};\node[below] at (3.2,2.2) {$\mathbf{U}^*_R$};\node[below] at (3.75,1.25) {$\mathbf{U}_R$};\node[below] at (1.6,2.2) {$\widehat{\mathbf{U}}$};
\end{tikzpicture}
x
t
xc
UL
U⇤L U⇤
R
UR
bUR
x
t
xc
UL
U⇤L U⇤
R
UR
bUL
bUL
x
t
xc
UL
U⇤L U⇤
R
UR
bURU⇤
L U⇤R
UR
x
t
xc
UL
\begin{tikzpicture}[scale=1] \draw[->] (0,0) -- (5,0); \draw[->] (0,0) -- (0,2.5); \draw (2.5,-.1) -- (2.5, .1); \draw[red, dashed, thick] (2.5,0) -- (2.7,2.5); \draw[red] (2.5,0) -- (4.2,2.5); \node[below] at (2.3,2.2) {$\mathbf{U}^*_L$}; \node[below] at (3.2,2.2) {$\mathbf{U}^*_R$}; \node[below] at (3.75,1.25) {$\mathbf{U}_R$}; \draw[->] (0,0) -- (5,0); \draw[->] (0,0) -- (0,2.5); \draw (2.5,-.1) -- (2.5, .1); \draw[red, dashed, thick] (2.5,0) -- (2.7,2.5); \draw[red] (2.5,0) -- (1.7,2.5); \node[below] at (5,-.1) {$x$};
10
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
The Riemann problem for the sensitivity equations
The sensitivity system writes:�1(U) = u� c,�2(U) = u,�3(U) = u+ c.
Eigenvalues:8<
:
@t
⇢a
+ @x
(⇢u)a
= S1,@t
(⇢u)a
+ @x
(⇢a
u2 + 2⇢uua
+ pa
) = S2,@t
(⇢E)a
+ @x
(ua
(⇢E + p) + u((⇢E)a
+ pa
)) = S3,
x
t
xc
Ua,L
U⇤a,L U⇤
a,R
Ua,R
bUa,L
\begin{tikzpicture}[scale=1] \draw[->] (0,0) -- (5,0); \draw[->] (0,0) -- (0,2.5); \draw (2.5,-.1) -- (2.5, .1); \draw[blue, very thick] (2.5,0) -- (0.9,2.5); \draw[blue] (2.5,0) -- (1.1,2.5); \draw[blue] (2.5,0) -- (1.3,2.5); \draw[blue] (2.5,0) -- (1.5,2.5); \draw[blue, very thick] (2.5,0) -- (1.7,2.5); \draw[red, dashed, thick] (2.5,0) -- (2.7,2.5); \draw[red] (2.5,0) -- (4.2,2.5); \node[below] at (5,-.1) {$x$}; \node[left] at (-.1,2.5) {$t$}; \node[below] at (2.5,-.1) {$x_c$}; \node[below] at (1.25,1.25) {$\mathbf{U}_{a,L}$}; \node[below] at (2.3,2.2) {$\mathbf{U}^*_{a,L}$}; \node[below] at (3.2,2.2) {$\mathbf{U}^*_{a,R}$}; \node[below] at (3.75,1.25) {$\mathbf{U}_{a,R}$}; \node[below] at (1.6,2.2) {$\widehat{\mathbf{U}}_{a,L}$};\end{tikzpicture}
x
t
xc
Ua,L
U⇤a,L
U⇤a,R
Ua,R
bUa,R
bUa,L
x
t
xc
Ua,L
U⇤a,L
U⇤a,R
Ua,R
bUa,RU⇤a,L U⇤
a,R
Ua,R
x
t
xc
Ua,L
\begin{tikzpicture}[scale=1] \draw[->] (0,0) -- (5,0); \draw[->] (0,0) -- (0,2.5); \draw (2.5,-.1) -- (2.5, .1); \draw[red, dashed, thick] (2.5,0) -- (2.7,2.5); \draw[red] (2.5,0) -- (4.2,2.5); \node[below] at (2.3,2.2) {$\mathbf{U}^*_{a,L}$}; \node[below] at (3.2,2.2) {$\mathbf{U}^*_{a,R}$}; \node[below] at (3.75,1.25) {$\mathbf{U}_{a,R}$}; \draw[->] (0,0) -- (5,0); \draw[->] (0,0) -- (0,2.5); \draw (2.5,-.1) -- (2.5, .1); \draw[red, dashed, thick] (2.5,0) -- (2.7,2.5); \draw[red] (2.5,0) -- (1.7,2.5); \node[below] at (5,-.1) {$x$};
11
ClassicalNumericalSchemes
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Classical numerical schemes
Step 0 : initial data discretisation
Step 1 : solution of a Riemann problem for each interface obtaining
Step 2 : average
Godunov-type schemes
Vn+1j
=1
�x
Zxj+1/2
xj�1/2
v(x, tn+1)dx
\mathbf{V}^{n+1}_j = \frac{1}{\Delta x} v(x, tn+1)xj�1/2
approximate Riemann solvers are used
Remark: the state and the sensitivity systems are not solved as a global system.
12
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Classical numerical schemes(@t
U+ @x
F(U) = 0
@t
Ua
+ @x
Fa
(U,Ua
) = S(U) S(U) =NsX
k=1
�a,k(U+k �U�
k )
Remark: HLL-type schemes cannot be used for the state, two intermediate star states are necessary to have a well-defined the source term for the sensitivity.
‣ First order Roe-type scheme
Approximate Riemann solver for the state
�ROE1 = u� c �ROE
2 = u �ROE3 = u+ c Roe-averaged eigenvalues
UR �UL =3X
k=1
↵iri decomposition along Roe-averaged eigenvectors
U⇤L = UL + ↵1r1 U⇤
R = UR � ↵3r3
13
[10] Bouchut, F. (2004). Nonlinear stability of finite Volume Methods for hyperbolic conservation laws: And Well-Balanced schemes for sources. Springer Science & Business Media.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Classical numerical schemes(@t
U+ @x
F(U) = 0
@t
Ua
+ @x
Fa
(U,Ua
) = S(U) S(U) =NsX
k=1
�a,k(U+k �U�
k )
Remark: HLL-type schemes cannot be used for the state, two intermediate star states are necessary to have a well-defined the source term for the sensitivity.
‣ First order Roe-type scheme‣ Second order Roe-type scheme
Time discretisation: two-step Runge-Kutta methodSpace discretisation: MUSCL-type scheme [10]
xxj�3/2 xj�1/2 xj+1/2 xj+3/2
Approximate Riemann solver for the state
13
[10] Bouchut, F. (2004). Nonlinear stability of finite Volume Methods for hyperbolic conservation laws: And Well-Balanced schemes for sources. Springer Science & Business Media.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Classical numerical schemes(@t
U+ @x
F(U) = 0
@t
Ua
+ @x
Fa
(U,Ua
) = S(U) S(U) =NsX
k=1
�a,k(U+k �U�
k )
Remark: HLL-type schemes cannot be used for the state, two intermediate star states are necessary to have a well-defined the source term for the sensitivity.
‣ First order Roe-type scheme‣ Second order Roe-type scheme
Time discretisation: two-step Runge-Kutta methodSpace discretisation: MUSCL-type scheme [10]
xxj�3/2 xj�1/2 xj+1/2 xj+3/2
Approximate Riemann solver for the state
13
[10] Bouchut, F. (2004). Nonlinear stability of finite Volume Methods for hyperbolic conservation laws: And Well-Balanced schemes for sources. Springer Science & Business Media.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Classical numerical schemes(@t
U+ @x
F(U) = 0
@t
Ua
+ @x
Fa
(U,Ua
) = S(U) S(U) =NsX
k=1
�a,k(U+k �U�
k )
Remark: HLL-type schemes cannot be used for the state, two intermediate star states are necessary to have a well-defined the source term for the sensitivity.
‣ First order Roe-type scheme‣ Second order Roe-type scheme
Time discretisation: two-step Runge-Kutta methodSpace discretisation: MUSCL-type scheme [10]
xxj�3/2 xj�1/2 xj+1/2 xj+3/2
Uj�5/4
Uj�3/4
Uj�1/4
Uj+1/4 Uj+3/4Uj+5/4
xxj�3/2 xj�1/2 xj+1/2 xj+3/2
Approximate Riemann solver for the state
13
[10] Bouchut, F. (2004). Nonlinear stability of finite Volume Methods for hyperbolic conservation laws: And Well-Balanced schemes for sources. Springer Science & Business Media.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Classical numerical schemes(@t
U+ @x
F(U) = 0
@t
Ua
+ @x
Fa
(U,Ua
) = S(U) S(U) =NsX
k=1
�a,k(U+k �U�
k )
Remark: HLL-type schemes cannot be used for the state, two intermediate star states are necessary to have a well-defined the source term for the sensitivity.
‣ First order Roe-type scheme‣ Second order Roe-type scheme
Time discretisation: two-step Runge-Kutta methodSpace discretisation: MUSCL-type scheme [10]
xxj�3/2 xj�1/2 xj+1/2 xj+3/2
Uj�5/4
Uj�3/4
Uj�1/4
Uj+1/4 Uj+3/4Uj+5/4
Approximate Riemann solver for the state
13
[10] Bouchut, F. (2004). Nonlinear stability of finite Volume Methods for hyperbolic conservation laws: And Well-Balanced schemes for sources. Springer Science & Business Media.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Classical numerical schemes
Approximate Riemann solvers for the sensitivity
‣ HLL-type scheme: simpler structure that the state solver. HLL consistency conditions yield:
U⇤a,L = Ua,L + ↵1,ar1 + ↵1r1,a U⇤
a,R = Ua,R � ↵3,ar3 � ↵3r3,a
U⇤a,j�1/2 =
1
�ROE3 � �ROE
1
⇣�ROE3 Un
a,j � �ROE1 Un
a,j�1 � Fa(Uj ,Ua,j) + Fa(Uj�1,Ua,j�1) + Sj�1/2
⌘
Sj�1/2 =@a�ROE1,j�1/2(U
⇤L,j�1/2 �Uj�1)d1,j�1/2
+@a�ROE2,j�1/2(U
⇤R,j�1/2 �U⇤
L,j�1/2)
+@a�ROE3,j�1/2(Uj �U⇤
R,j�1/2)d3,j�1/2
‣ HLLC-type scheme: same structure as the state. HLL consistency conditions + Rankine-Hugoniot conditions. Equivalent to:
14
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Classical numerical schemes
⇢pL(x, T )
0 0.2 0.4 0.6 0.8 1
�0.5
0
0.5
1
1.5
2
Numerical S = 0
Exact
15
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Classical numerical schemes
⇢pL(x, T )
15
0 0.2 0.4 0.6 0.8 1
�0.4
�0.2
0
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Classical numerical schemes
⇢pL(x, T )
0 0.2 0.4 0.6 0.8 1
�0.4
�0.2
0
Exact
1st order - HLLC
2nd order - HLLC
16
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Isolated shock for the p-system
17
0 0.2 0.4 0.6 0.8 1
0.2
0.3
0.4
0.5 Exact
Godunov
Roe I
Roe II
0 0.2 0.4 0.6 0.8 1
�3
�2.5
�2
�1.5
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
�10
�8
�6
�4
�2
0
⌧(x, T )
⌧a(x, T ) ua(x, T )
u(x, T )
NumericalScheme
without diffusion
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Scheme without numerical diffusion
Step 0 : initial data discretisation
t
xxj�1/2
Uj�1 Uj
xj+1/2
Uj+1
18
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Scheme without numerical diffusion
Step 0 : initial data discretisation
Step 1 : solution of the Riemann problems, one for each interface
t
xxj�1/2
Uj�1 Uj
xj+1/2
Uj+1
t
xxj�1/2
Uj�1 Uj
xj+1/2
Uj+1
18
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Scheme without numerical diffusion
Step 0 : initial data discretisation
Step 1 : solution of the Riemann problems, one for each interface
Step 2 : average
t
xxj�1/2
Uj�1 Uj
xj+1/2
Uj+1
t
xxj�1/2
Uj�1 Uj
xj+1/2
Uj+1
18
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Scheme without numerical diffusion
Step 0 : initial data discretisation
Step 1 : solution of the Riemann problems, one for each interface
Step 2 : average
t
xxj�1/2
Uj�1 Uj
xj+1/2
Uj+1
t
xxj�1/2
Uj�1 Uj
xj+1/2
Uj+1
18
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Step 0 : initial data discretisation
Step 1 : solution of the Riemann problems, one for each interface
Step 2 : definition of a staggered mesh on which the average is performed [11]
Scheme without numerical diffusion
t
xxj�1/2
Uj�1 Uj
xj+1/2
Uj+1
t
xxj�1/2
Uj�1 Uj
xj+1/2
Uj+1
18
[11] Chalons, C., & Goatin, P. (2008). Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling. Interfaces and Free Boundaries, 10(2), 197-221.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Step 0 : initial data discretisation
Step 1 : solution of the Riemann problems, one for each interface
Step 2 : definition of a staggered mesh on which the average is performed [11]
xj�1/2 = xj�1/2 + �j�1/2�t
Scheme without numerical diffusion
t
x
xj�1/2 xj+1/2
xj�1/2
Uj�1 Uj
xj+1/2
Uj+1
t
xxj�1/2
Uj�1 Uj
xj+1/2
Uj+1
t
xxj�1/2
Uj�1 Uj
xj+1/2
Uj+1
18
[11] Chalons, C., & Goatin, P. (2008). Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling. Interfaces and Free Boundaries, 10(2), 197-221.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Step 0 : initial data discretisation
Step 1 : solution of the Riemann problems, one for each interface
Step 2 : definition of a staggered mesh on which the average is performed
Step 3 : projection on the initial mesh [12]\begin{tikzpicture}% cn\draw[thick, blue] (0, 2) -- (6, 2);\draw[thick, dashed, blue] (-.5, 2) -- (0, 2);\draw[thick, dashed, blue] (6, 2) -- (6.5, 2);\foreach \i in {0,2,4,6} \draw [blue](\i, 1.9) -- (\i, 2.1);\node [above] at (1, 2.1) {${\color{blue}\mathbf{U}_{j-1}}$};\node [above] at (3, 2.1) {${\color{blue}\mathbf{U}_j}$};\node [above] at (5, 2.1) {${\color{blue}\mathbf{U}_{j+1}}$};% cd\def\k{1.7};\draw[thick, red] (0, \k) -- (6, \k);\draw[thick, dashed, red] (-.5, \k) -- (0, \k);\draw[thick, dashed, red] (6,\k) -- (6.5, \k);\draw [red](.15, \k-.1) -- (0.15, \k+.1);\draw [red](2.4, \k-.1) -- (2.4, \k+.1);\draw [red](3.75, \k-.1) -- (3.75, \k+.1);\draw [red](6, \k-.1) -- (6, \k+.1);\node[below] at (1.275, \k-.1) {${\color{red}\overline{\mathbf{U}}_{j-1}}$};\node[below] at (3.075, \k-.1) {${\color{red}\overline{\mathbf{U}}_{j}}$};\node[below] at (4.875, \k-.1) {${\color{red}\overline{\mathbf{U}}_{j+1}}$};
Uj�1 Uj Uj+1
Uj�1 Uj Uj+1
Scheme without numerical diffusion
Uj
=
8><
>:
Uj�1 if ↵ 2
�0, �t
�x
max(�j�1/2, 0)
�,
Uj
if ↵ 2⇥�t
�x
max(�j�1/2, 0), 1 +
�t
�x
min(�j+1/2, 0)
�,
Uj+1 if ↵ 2
⇥1 +
�t
�x
min(�j+1/2, 0), 1
�.
↵ ⇠ U([0, 1])
18
[12] Glimm, J. (1965). Solutions in the large for nonlinear hyperbolic systems of equations. Communications on pure and applied mathematics, 18(4), 697-715.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Numerical results
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
ExactFirst order
First order AD
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
�0.4
�0.2
0
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
u(x, T )⇢(x, T ) p(x, T )
ppL(x, T )upL(x, T )⇢pL(x, T )
19
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Numerical results
0.64 0.65 0.66 0.67 0.68 0.69 0.7
0
0.2
0.4
0.6
0.8
1
0.64 0.65 0.66 0.67 0.68 0.69 0.7
0.15
0.2
0.25
ExactFirst order
First order AD
0.64 0.65 0.66 0.67 0.68 0.69 0.7
0.1
0.15
0.2
0.25
0.3
0.4 0.45 0.5 0.55 0.6 0.65 0.7
0
0.2
0.4
0.6
0.4 0.45 0.5 0.55 0.6 0.65 0.7
0
0.1
0.2
0.3
0.4
0.5
u(x, T )⇢(x, T ) p(x, T )
ppL(x, T )upL(x, T )⇢pL(x, T )
19
0.6 0.62 0.64 0.66 0.68 0.7
0
0.05
0.1
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Numerical results
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
ExactFirst order ADSecond order AD
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
�0.4
�0.2
0
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
u(x, T )⇢(x, T ) p(x, T )
ppL(x, T )upL(x, T )⇢pL(x, T )
20
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Numerical results
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
ExactFirst order ADSecond order AD
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
�0.4
�0.2
0
ExactHLLCHLL
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
u(x, T )⇢(x, T ) p(x, T )
ppL(x, T )upL(x, T )⇢pL(x, T )
21
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Numerical results
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
ExactFirst order ADSecond order AD
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.46 0.48 0.5 0.52 0.54
�0.12
�0.1
�8 · 10�2
�6 · 10�2
�4 · 10�2
�2 · 10�2
ExactHLLCHLL
0.46 0.48 0.5 0.52 0.54
0.5
0.55
0.6
0.65
0.46 0.48 0.5 0.52 0.54
0.25
0.3
0.35
u(x, T )⇢(x, T ) p(x, T )
ppL(x, T )upL(x, T )⇢pL(x, T )
22
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Convergence
10�5 10�4 10�3 10�2
10�4
10�3
10�2 Roe IRoe II
Roe I ADRoe II AD
10�5 10�4 10�3 10�2
10�5
10�4
10�3
10�2
Roe IRoe II
Roe I ADRoe II AD
10�5 10�4 10�3 10�2
10�5
10�4
10�3
10�2 Roe IRoe II
Roe I ADRoe II AD
10�5 10�4 10�3 10�2
10�3
10�2
Roe IRoe II
Roe I ADRoe II AD
10�5 10�4 10�3 10�2
10�3
10�2
Roe IRoe II
Roe I ADRoe II AD
10�5 10�4 10�3 10�2
10�3
10�2
Roe IRoe II
Roe I ADRoe II AD
k⇢expL(x, T )� ⇢
pL(x, T )kL1(0,1) kuex
pL(x, T )� u
pL(x, T )kL1(0,1) kpexpL(x, T )� p
pL(x, T )kL1(0,1)
kpex(x, T )� p(x, T )kL
1(0,1)kuex(x, T )� u(x, T )kL
1(0,1)k⇢ex(x, T )� ⇢(x, T )kL
1(0,1)
23
Applications
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Uncertainty Quantification
24
Let a be a random vector, with the following average and variance:
µa =
2
64µa1
.
.
.
µaM
3
75 , �a =
2
6664
�2a1
cov(a1, a2) . . . cov(a1, aM )
cov(a1, a2) �2a2
. . . cov(a2, aM )
.
.
.
.
.
.
.
.
.
cov(a1, aM ) . . . �2aM
3
7775,
Aim: determine a confidence interval CIX = [µX � �X , µX + �X ]
Monte Carlo approach: N samples of the state
�2X =
1
N � 1
NX
k=1
|µX �Xk|2µX =1
N
NX
k=1
Xk
Xk
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Uncertainty Quantification
25
Sensitivity approach: let us consider the following first order Taylor expansion
X(a) = X(µa) +MX
i=1
(ai � µai)Xai(µa) + o(kak2).
µX = E[X(a)] = X(µa) +MX
i=1
Xai(µa)E[ai � µai ] = X(µa),
Then, computing, the average one has:
And for the variance:
�2X = E[(X(a)� µX)2] = E
2
4
MX
i=1
Xai(µa)(ai � µai)
!23
5 =
=MX
i=1
X2ai(µa)E[(ai � µai)
2] +MX
i,j=1i 6=j
Xai(µa)Xaj (µa)E[(ai � µai)(aj � µaj )].
Therefore, we have the following first order estimates:
µX = X(µa), �2X =
MX
i=1
X2ai�2ai
+
MX
i,j=1i 6=j
XaiXajcov(ai, aj).
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Uncertainty Quantification
26
Test case:
Riemann problem with uncertain parameters: a = (⇢L, ⇢R, uL, uR, pL, pR)t
µa = (1, 0.125, 0, 0, 1, 0.1)t, �a = diag(0.001, 0.000125, 0.0001, 0.0001, 0.001, 0.0001).
with the following average and covariance matrix:
�2X =
6X
i=1
X2ai�2ai
Since the covariance matrix is diagonal, the previous estimate is simplified:
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Uncertainty Quantification
27
Monte Carlo method:
Sensitivity method without correction:
⇢
⇢
u
u
p
p
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
�0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
�0.5
0
0.5
1
0 0.2 0.4 0.6 0.8 1�6
�4
�2
0
2
4
6
0 0.2 0.4 0.6 0.8 1
�1
0
1
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Uncertainty Quantification
27
Monte Carlo method:
Sensitivity method without correction:
⇢
⇢
u
u
p
p
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
�0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1�0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Uncertainty Quantification
28
Monte Carlo method:
Sensitivity method with correction (diffusive method):
⇢
⇢
u
u
p
p
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
�0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Uncertainty Quantification
29
Monte Carlo method:
Sensitivity method with correction (AD method):
⇢
⇢
u
u
p
p
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
�0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Optimization
8<
:
@t
(h⇢) + @x
(h⇢u) = 0,@t
(h⇢u) + @x
(h⇢u2 + p) = p@x
h,@t
(h⇢E) + @x
(hu(⇢E + p)) = 0,
The quasi-1D Euler equations are:
J(U) =1
2kp� p⇤k2L2Cost functional:
mina2A
J(U)Optimization problem: subject to (1).
(1)
+b.c.
30
\begin{tikzpicture}
yticklabels={$1$,$2$}]\addplot [samples=200,smooth, very thick, blue] {2-sin(deg((x-0.2)*pi/0.2-pi*0.5))*sin(deg((x-0.2)*pi/0.2-pi*0.5))*(x>0.1)*(x<0.3)};\addplot[|-|] coordinates {(0.1,1.5) (0.3,1.5)};\node[below] at (axis cs: 0.2,1.5) {$\ell$};\end{axis}
xc
1
2
`
h(x)
Parameters: a = (xc, `)t
Target pressure: p
⇤ = p(xc = 0.5, ` = 0.5)
Gradient: raJ(U) =
(p� p⇤, p
xc)L2
(p� p⇤, p`
)L
2
�
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Optimization
31
\begin{tikzpicture}
yticklabels={$1$,$2$}]\addplot [samples=200,smooth, very thick, blue] {2-sin(deg((x-0.2)*pi/0.2-pi*0.5))*sin(deg((x-0.2)*pi/0.2-pi*0.5))*(x>0.1)*(x<0.3)};\addplot[|-|] coordinates {(0.1,1.5) (0.3,1.5)};\node[below] at (axis cs: 0.2,1.5) {$\ell$};\end{axis}
0 0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
p
⇤(x)
J(xc, `)
Boundary conditions:‣ inlet: enthalpy and total pressure ‣ outlet: pressure
HL ptot,L
pR
8>>>><
>>>>:
H = E +p
⇢,
p = (� � 1)
✓⇢E � 1
2⇢u2
◆,
ptot
= p+1
2⇢u2.
These b.c. provide a discontinuous solution [13] 8a 2 A.
[13] Giles, M. B., & Pierce, N. A. (2001). Analytic adjoint solutions for the quasi-one-dimensional Euler equations. Journal of Fluid Mechanics, 426, 327-345.
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Optimization
�0.2 0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
With correction
Without correction
xc
32
`
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Optimization
0.35 0.4 0.45 0.5 0.55 0.60.35
0.4
0.45
0.5
0.55With correction
Without correction
xc
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Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Optimization
log(krJk)log(J)
0 10 20 30
�6
�4
�2
Without correction
With correction
0 10 20 30
�2
�1
0
Without correction
With correction
iterations iterations
33
Camilla Fiorini Sensitivity Analysis for nonlinear hyperbolic PDEs / 34
Conclusion and perspectives
‣ Extension to non-classical shocks ‣ Effects of the numerical diffusion for the applications ‣ Extension to 2D
‣ We defined a sensitivity system valid in case of discontinuous state ‣ The correction term is well defined ‣ The correction term is important in applications
Conclusion:
Future development:
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Thank youfor your attention!